Problem 3
Consider the system \(\mathscr{S}\) of all subsets of the real line containing only finitely many points, and let the measure \(\mu(A)\) of a set \(A \in \mathscr{S}\) be defined as the number of points in \(A\). Prove that a) \(\mathscr{S}\) is a ring without a unit; b) \(\mu\) is not o-finite.
Problem 10
Prove that the integral $$ \int_{0}^{1} \frac{1}{x} \sin \frac{1}{x} d x $$ exists as an improper Riemann integral, but not as a Lebesgue integral.
Problem 11
Suppose \(\mathbf{f}\) is Riemann integrable over an infinite interval (such an integral can exist only in the improper sense). Prove that \(\mathbf{f}\) is Lebesgue integrable over the same interval if and only if the improper integral converges absolutely. Comment. For example, the function. $$ \mathrm{f}(x)=\frac{\sin \mathrm{x}}{x} $$ is not Lebesgue integrable over \((-\infty, \infty)\), since $$ \int_{-\infty}^{\infty}\left|\frac{\sin x}{x}\right| d x=\infty $$ On the other hand, \(\mathbf{f}\) has an improper Riemann integral equal to $$ \int_{-\infty}^{\infty} \frac{\sin x}{x}=\pi $$
